# What is the time dilation?

Works out special-relativity time dilation: how long a time on a moving clock lasts for an observer at rest, Δt = γΔτ, with the Lorentz factor γ, from a speed in % of light speed or in m/s, km/h or mph.

- Page: https://www.acalculator.org/physics/time-dilation-calculator
- JSON spec: https://www.acalculator.org/physics/time-dilation-calculator.json
- Version: 05a974b0d65d

## Default answer

Example with the default inputs (Speed as % of light speed, Speed (% of c) 80%, Time on the moving clock 1 yr): At 80% of the speed of light, 1 yr on the moving clock lasts 1.666666667 yr for an observer at rest.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| by | Speed as | Type the speed as a percent of the speed of light, or as a speed in m/s, km/h or mph. |
| b | Speed (% of c) | The speed of the moving clock as a percent of the speed of light, from 0 up to (not including) 100. |
| v | Speed | The speed of the moving clock, below the speed of light (299,792,458 m/s). |
| t | Time on the moving clock | The proper time Δτ: how long the event lasts on a clock that moves with it. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| dilated | Time for an observer at rest | The dilated time Δt = γ × Δτ, in the unit of the time you typed. |
| gamma | Lorentz factor (γ) | γ = 1 ÷ √(1 − v²/c²), always 1 or more. |
| extra | Extra time | How much longer the event lasts for the observer at rest: Δt − Δτ, in the unit of the time you typed. |
| pct | Speed (% of c) | The speed as a percent of the speed of light. |
| mps | Speed (m/s) | The speed in metres per second (c = 299,792,458 m/s). |

## Method

Δt = γ × Δτ, γ = 1 ÷ √(1 − v²/c²), c = 299,792,458 m/s.

## Assumptions

- Special relativity: the clock moves at a constant speed relative to the observer; gravity (general relativity) is left out.
- The speed of light is exactly 299,792,458 m/s; 1 mph = 0.44704 m/s; 1 km/h = 1/3.6 m/s; a year is 365 days.

## Worked examples

1. by = percent, b = 95%, t = 0.000002 gives dilated = 0.000007, gamma = 3.202563. Source: OpenStax, University Physics Volume 3, §5.3 Time Dilation (Δt = Δτ ÷ √(1 − v²/c²)), https://openstax.org/books/university-physics-volume-3/pages/5-3-time-dilation (Example 5.3: Δt = 7.05 μs).
2. by = percent, b = 80%, t = 31,536,000 gives dilated = 52,560,000, gamma = 1.666667, extra = 21,024,000. Source: OpenStax, University Physics Volume 3, §5.3 Time Dilation (Δt = Δτ ÷ √(1 − v²/c²)), https://openstax.org/books/university-physics-volume-3/pages/5-3-time-dilation.
3. by = speed, v = 5,830, t = 1 gives dilated = 1, extra = 0. Source: OpenStax, University Physics Volume 3, §5.3 Time Dilation (Δt = Δτ ÷ √(1 − v²/c²)), https://openstax.org/books/university-physics-volume-3/pages/5-3-time-dilation (Example 5.1: Δt = 1 s + 1.89 × 10⁻¹⁰ s).
4. by = percent, b = 60%, t = 3,600 gives dilated = 4,500, gamma = 1.25, mps = 179,875,474.8. Source: OpenStax, University Physics Volume 3, §5.3 Time Dilation (Δt = Δτ ÷ √(1 − v²/c²)), https://openstax.org/books/university-physics-volume-3/pages/5-3-time-dilation.

## FAQ

### What is the time dilation formula?

Δt = γ × Δτ, where Δτ is the time on the moving clock (the proper time), Δt is the time an observer at rest measures, and γ = 1 ÷ √(1 − v²/c²) is the Lorentz factor. At 80% of light speed, γ = 1 ÷ √(1 − 0.64) = 1 ÷ 0.6 = 5/3, so 1 year on the moving clock is 1.667 years for the observer.

### What is the Lorentz factor?

The Lorentz factor γ = 1 ÷ √(1 − v²/c²) says how much a moving clock runs slow. It is 1 at rest, 1.25 at 60% of light speed, 3.2 at 95%, and grows without limit as the speed nears c.

### Why does a muon reach the ground?

A muon lives 2.20 μs on its own clock. At 0.950c, γ = 3.20, so in the Earth frame it lives 7.05 μs and travels about three times farther than it could without time dilation (OpenStax Example 5.3).

### Is there time dilation at everyday speeds?

Yes, but it is tiny. A craft at 5,830 m/s gains only 1.89 × 10⁻¹⁰ s per second (OpenStax Example 5.1). The page works out that small difference without rounding it away.

### Can anything reach the speed of light?

No object with mass can. At v = c the formula divides by zero, so the page takes speeds from 0 up to, but not including, 100% of c.

### Does this include gravitational time dilation?

No. This page uses special relativity only: a clock moving at a constant speed. Clocks deeper in a gravity field also run slow (general relativity), which GPS satellites must correct for, but that is a different formula.

## Sources

- OpenStax, University Physics Volume 3, §5.3 Time Dilation (Δt = Δτ ÷ √(1 − v²/c²); Examples 5.1 and 5.3). https://openstax.org/books/university-physics-volume-3/pages/5-3-time-dilation (retrieved 2026-10-03)
- NIST, CODATA value: speed of light in vacuum, c = 299,792,458 m/s (exact). https://physics.nist.gov/cgi-bin/cuu/Value?c (retrieved 2026-10-03)